Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

Abstract

We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions $$\mathbb {S}^{d-1}$$ S d - 1 . We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is $$O(\min \{k,n-k\}n^{d-1})$$ O ( min { k , n - k } n d - 1 ) , which is tight for $$n-k=O(1)$$ n - k = O ( 1 ) . This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its $$(d-1)$$ ( d - 1 ) -skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of  $$n \ge 2$$ n ≥ 2 lines in general position has exactly $$n(n-1)$$ n ( n - 1 ) three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in $$O(n^{d-1} \alpha (n))$$ O ( n d - 1 α ( n ) ) time, for $$d\ge 4$$ d ≥ 4 , while if $$d=3$$ d = 3 , the time drops to worst-case optimal $$\Theta (n^2)$$ Θ ( n 2 ) . We extend the obtained results to bounded polyhedra and clusters of points as sites.